Normal subgroups of mapping class groups and the metaconjecture of Ivanov

Tara Brendle, Dan Margalit

Published 2017-10-24Version 1

We consider a large class of normal subgroups of the mapping class group of a closed surface, and we show that all members of this class have automorphism group and abstract commensurator group isomorphic to the mapping class group. A normal subgroup lies in this class if, for example, it contains a pure element whose support is contained non-peripherally in a subsurface with one boundary component and genus less than one third that of the whole surface. The proof relies on another theorem, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the mapping class group. These results resolve in part the metaconjecture of N.V. Ivanov, which asserts that any "sufficiently rich" object associated to a surface has automorphism group isomorphic to the extended mapping class group.